Scattered photons which carry false information, and make a large contribution to noise in emission tomography, can constitute as much as 50% of the total events collected depending on the construction of the imaging instrument. This contribution is not uniform in the majority of situations and will necessarily lead to erroneous interpretation of data when quantitative results are sought. Whereas some simple computational schemes have provided methods for removing the majority of scatter background in positron emission tomographs for head imaging, these methods are not applicable to the general problem of variable attenuation coefficient nor to recent tomographic designs which seek to minimize interplane shielding in order to collect a greater solid angle of coincidence data. Neither do existing methods compensate off-plane sources such as spleen and liver activity which can contribute serious data distortion in transverse section imaging of the thorax. We propose to remove the scatter contribution and to optimize design of single and positron emission tomographic shielding and detector configurations. Our methods rely on rigorous Monte Carlo simulations which use the Klein-Nishina collision cross sections in a flexible computing architecture. An essential tool which has been developed for this application allows for variable attenuation and tracks the trajectories of photons in three dimensions. Using these Monte Carlo methods the scatter background will be characterized for general cases of distributed sources and scattering media. The expected benefits of this research are twofold: first, an efficient algorithm will be developed for removal of in-plane and off-plane scatter contributions by iterative schemes or deconvolution methods; second, the results of these calculations will be used to examine various shielding and detector configurations for improved quantitation in high-resolution positron tomography with multilavered tomographic designs.